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Mirrors > Home > HSE Home > Th. List > bdophsi | Structured version Visualization version GIF version |
Description: The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | ⊢ 𝑆 ∈ BndLinOp |
nmoptri.2 | ⊢ 𝑇 ∈ BndLinOp |
Ref | Expression |
---|---|
bdophsi | ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 ⊢ 𝑆 ∈ BndLinOp | |
2 | bdopln 31691 | . . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝑆 ∈ LinOp |
4 | nmoptri.2 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
5 | bdopln 31691 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇 ∈ LinOp |
7 | 3, 6 | lnophsi 31831 | . 2 ⊢ (𝑆 +op 𝑇) ∈ LinOp |
8 | bdopf 31692 | . . . . . 6 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ 𝑆: ℋ⟶ ℋ |
10 | bdopf 31692 | . . . . . 6 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
11 | 4, 10 | ax-mp 5 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ |
12 | 9, 11 | hoaddcli 31598 | . . . 4 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
13 | nmopxr 31696 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → (normop‘(𝑆 +op 𝑇)) ∈ ℝ*) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ* |
15 | nmopre 31700 | . . . . 5 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ) | |
16 | 1, 15 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑆) ∈ ℝ |
17 | nmopre 31700 | . . . . 5 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
18 | 4, 17 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑇) ∈ ℝ |
19 | 16, 18 | readdcli 11267 | . . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ |
20 | nmopgtmnf 31698 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝑆 +op 𝑇))) | |
21 | 12, 20 | ax-mp 5 | . . 3 ⊢ -∞ < (normop‘(𝑆 +op 𝑇)) |
22 | 1, 4 | nmoptrii 31924 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) |
23 | xrre 13188 | . . 3 ⊢ ((((normop‘(𝑆 +op 𝑇)) ∈ ℝ* ∧ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝑆 +op 𝑇)) ∧ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)))) → (normop‘(𝑆 +op 𝑇)) ∈ ℝ) | |
24 | 14, 19, 21, 22, 23 | mp4an 691 | . 2 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ |
25 | elbdop2 31701 | . 2 ⊢ ((𝑆 +op 𝑇) ∈ BndLinOp ↔ ((𝑆 +op 𝑇) ∈ LinOp ∧ (normop‘(𝑆 +op 𝑇)) ∈ ℝ)) | |
26 | 7, 24, 25 | mpbir2an 709 | 1 ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 class class class wbr 5152 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 + caddc 11149 -∞cmnf 11284 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 ℋchba 30749 +op chos 30768 normopcnop 30775 LinOpclo 30777 BndLinOpcbo 30778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-hilex 30829 ax-hfvadd 30830 ax-hvcom 30831 ax-hvass 30832 ax-hv0cl 30833 ax-hvaddid 30834 ax-hfvmul 30835 ax-hvmulid 30836 ax-hvmulass 30837 ax-hvdistr1 30838 ax-hvdistr2 30839 ax-hvmul0 30840 ax-hfi 30909 ax-his1 30912 ax-his2 30913 ax-his3 30914 ax-his4 30915 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-grpo 30323 df-gid 30324 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-nmcv 30430 df-hnorm 30798 df-hba 30799 df-hvsub 30801 df-hosum 31560 df-nmop 31669 df-lnop 31671 df-bdop 31672 |
This theorem is referenced by: bdophdi 31927 nmoptri2i 31929 unierri 31934 |
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